Local nonlinear balancing for structural information with an application in super collider particle dynamics

The information obtained by local balancing of a nonlinear system already provides useful information about the dominant dynamics of the system and the topology of the state space. For local balancing, two Riemannian metrics are specified: one models the local reachability properties and one models the local observability properties. In general these are incompatible, inducing a different global topology. Locally, it is possible to match these up, and local balancing at a point P corresponds to bending and reshaping the induced manifolds without tearing so that near P there is a snug fit between the manifolds. Unlike the linear case, sensitivity and reduced modeling are local concepts, and lead at best to a hybrid reduced model with modes of different dimension. These concepts are illustrated in the nonlinear reachability and observability maps for the dynamics of bunching in a particle accelerator. Considering the recent outburst of interest in flocking, this example in ultimate flocking may be exciting and inspiring.